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Mirrors > Home > MPE Home > Th. List > cnmpt1plusg | Structured version Visualization version GIF version |
Description: Continuity of the group sum; analogue of cnmpt12f 21469 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
cnmpt1plusg.p | ⊢ + = (+g‘𝐺) |
cnmpt1plusg.g | ⊢ (𝜑 → 𝐺 ∈ TopMnd) |
cnmpt1plusg.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) |
cnmpt1plusg.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) |
cnmpt1plusg.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) |
Ref | Expression |
---|---|
cnmpt1plusg | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1plusg.k | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) | |
2 | cnmpt1plusg.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TopMnd) | |
3 | tgpcn.j | . . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝐺) | |
4 | eqid 2622 | . . . . . . . . 9 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 3, 4 | tmdtopon 21885 | . . . . . . . 8 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
6 | 2, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
7 | cnmpt1plusg.a | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) | |
8 | cnf2 21053 | . . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺)) | |
9 | 1, 6, 7, 8 | syl3anc 1326 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺)) |
10 | eqid 2622 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | |
11 | 10 | fmpt 6381 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ (Base‘𝐺) ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺)) |
12 | 9, 11 | sylibr 224 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ (Base‘𝐺)) |
13 | 12 | r19.21bi 2932 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐺)) |
14 | cnmpt1plusg.b | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) | |
15 | cnf2 21053 | . . . . . . 7 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺)) | |
16 | 1, 6, 14, 15 | syl3anc 1326 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺)) |
17 | eqid 2622 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵) | |
18 | 17 | fmpt 6381 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ (Base‘𝐺) ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺)) |
19 | 16, 18 | sylibr 224 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ (Base‘𝐺)) |
20 | 19 | r19.21bi 2932 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝐺)) |
21 | cnmpt1plusg.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
22 | eqid 2622 | . . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺) | |
23 | 4, 21, 22 | plusfval 17248 | . . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) |
24 | 13, 20, 23 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵)) |
25 | 24 | mpteq2dva 4744 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵))) |
26 | 3, 22 | tmdcn 21887 | . . . 4 ⊢ (𝐺 ∈ TopMnd → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
27 | 2, 26 | syl 17 | . . 3 ⊢ (𝜑 → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
28 | 1, 7, 14, 27 | cnmpt12f 21469 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ (𝐾 Cn 𝐽)) |
29 | 25, 28 | eqeltrrd 2702 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 TopOpenctopn 16082 +𝑓cplusf 17239 TopOnctopon 20715 Cn ccn 21028 ×t ctx 21363 TopMndctmd 21874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-topgen 16104 df-plusf 17241 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-tx 21365 df-tmd 21876 |
This theorem is referenced by: tmdmulg 21896 tmdgsum 21899 tmdlactcn 21906 clsnsg 21913 tgpt0 21922 cnmpt1mulr 21985 |
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